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    <title>projspec</title>
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    <center>Scilab Function</center>
    <div align="right">Last update : April 1993</div>
    <p>
      <b>projspec</b> -  spectral operators</p>
    <h3>
      <font color="blue">Calling Sequence</font>
    </h3>
    <dl>
      <dd>
        <tt>[S,P,D,i]=projspec(A)  </tt>
      </dd>
    </dl>
    <h3>
      <font color="blue">Parameters</font>
    </h3>
    <ul>
      <li>
        <tt>
          <b>A</b>
        </tt>: square matrix</li>
      <li>
        <tt>
          <b>S, P, D</b>
        </tt>: square matrices</li>
      <li>
        <tt>
          <b>i</b>
        </tt>: integer (index of the zero eigenvalue of <tt>
          <b>A</b>
        </tt>).</li>
    </ul>
    <h3>
      <font color="blue">Description</font>
    </h3>
    <p>
    Spectral characteristics of <tt>
        <b>A</b>
      </tt> at 0.</p>
    <p>
      <tt>
        <b>S</b>
      </tt> = reduced resolvent at 0 (<tt>
        <b>S</b>
      </tt> = -Drazin_inverse(<tt>
        <b>A</b>
      </tt>)).</p>
    <p>
      <tt>
        <b>P</b>
      </tt> = spectral projection at 0.</p>
    <p>
      <tt>
        <b>D</b>
      </tt> = nilpotent operator at 0.</p>
    <p>
      <tt>
        <b>index</b>
      </tt> = index of the 0 eigenvalue.</p>
    <p>
    One has <tt>
        <b>(s*eye()-A)^(-1) = D^(i-1)/s^i +... + D/s^2 + P/s - S - s*S^2 -...</b>
      </tt>
    around the singularity s=0.</p>
    <h3>
      <font color="blue">Examples</font>
    </h3>
    <pre>

deff('j=jdrn(n)','j=zeros(n,n);for k=1:n-1;j(k,k+1)=1;end')
A=sysdiag(jdrn(3),jdrn(2),rand(2,2));X=rand(7,7);
A=X*A*inv(X);
[S,P,D,index]=projspec(A);
index   //size of J-block
trace(P)  //sum of dimensions of J-blocks
A*S-(eye()-P)
norm(D^index,1)
 
  </pre>
    <h3>
      <font color="blue">See Also</font>
    </h3>
    <p>
      <a href="coff.htm">
        <tt>
          <b>coff</b>
        </tt>
      </a>,&nbsp;&nbsp;</p>
    <h3>
      <font color="blue">Author</font>
    </h3>
    <p>F. D.; ;   </p>
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